Research Journal of Applied Sciences, Engineering and Technology 6(15): 2757-2763, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: December 28, 2012
Accepted: February 08, 2013
Published: August 20, 2013
A Preliminary Study on Material Utilization of Stiffened Cylindrical Shells
1, 2
Li Zhu and 1Xu Bai
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
2
Equipment Procurement Center, Equipment Department of Navy, Beijing 100071, China
1
Abstract: The stiffened cylindrical shell is used for legs of offshore platforms and pressure hull of submersibles
frequently. The existing design methods only focus on the strength and stability of structural components. The aim
of this study is to present a new criterion for design of stiffened cylindrical shells. As a reference, an un-stiffened
cylindrical shell is taken into consideration. By fixing the main scale value of structure, changing the shell thickness,
the dimensions and numbers of stiffeners, cylindrical shells with circumferential ribs are created. Then, the buckling
analysis on the family of shells, which is the revolution of constant mass, is carried out by FEM. Based on moving
least square method with interpolation conditions, the fitting surface of polynomial function is got by MATLAB
simulation. Results of polynomial show the relationship between the material utilization of structure and the critical
load under uniform external pressure.
Keywords: Buckling analysis, material utilization, shell structural design, stiffened cylindrical
INTRODUCTION
Stiffened cylindrical shells are widely used in
offshore engineering, such as legs of offshore platforms
and submarine pressure hulls. The design problems of
these kinds of structures have been a subject over the last
decades and are still vividly and broadly investigated
today. It is because there are still many unsolved
problems. Nowadays, issues have concerning the field of
cylindrical shell structure, as in the book of Chen (2001)
and the paper of Young-Shin (2009).
There are many papers considered the optimization
design of stiffened cylindrical shell, for example, the
optimum design problem of cylindrical shell under
arbitrary asymmetrical boundary conditions and with
uniform distributed radial pressure is reported by Lang
and Yue (2002). Then, in order to decrease the structural
weight and increase the effective payload, the stiffened
cylindrical shell of working platform in deep sea was
optimized by Gao-feng et al. (2010). Relatively new
results based on multi-objective optimization algorithm
for ring stiffened cylindrical shells were contained in the
paper of Bagheri et al. (2011). The main requirements of
modern engineering structures are that they should be
safe for load-carrying capacity, fit for production and be
economical. However, very few papers related to that.
The research on minimum cost of a welded orthogonally
stiffened cylindrical shell was presented by Jarmai et al.
(2006), in which the cost function and the functions
specifying the constraints are shown. And minimization
of the weight of ribbed cylindrical shells is considered by
Siryus (2011). According to these achievements, the
ideal design way of stiffened cylindrical shell seems to
be a method which consists with material utilization and
structure utilization besides strength and stability.
Fig. 1: The geometry of a cylindrical shell
The aim of this study is to expound material
utilization in structural design of stiffened cylindrical
shell. A procedure for a whole family of shells which
evolved from keeping constant mass is working out. The
un-stiffened cylindrical shell is taken into consideration
as reference. Figure 1 presents the geometry of a
cylindrical shell, the value of the main scale of structure,
radius of shell R and axial length L, are fixed. Then, a
family of stiffened cylindrical shells with circumferential
ribs is created by changing the shell thickness, the
dimensions and numbers of stiffeners. Furthermore,
comparing to other factors, the stability of cylindrical
shell is especially important in cylindrical shell of ship
and offshore engineering. The effect of material
utilization on critical load is the mainly consideration in
this study. ANSYS code, one of the Finite Element
Methods (FEM) software, is utilized to calculate critical
load by buckling analysis. Finally, material utilization
will be got from analyzing the FEM results.
Corresponding Author: Xu Bai, College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China
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Res. J. Appl. Sci. Eng. Technol., 6(15): 2757-2763, 2013
STIFFENED CYLINDRICAL SHELLS
WITH CONSTANT MASS
Firstly, create a family of stiffened cylindrical shells
which evolved from keeping constant mass. As a
reference, a column of ends both fixed is constructed and
loaded by uniform external pressure. Then, square crosssection ribs are welded inside of the shell by
circumferential fillet welds.
Hence, mass of the stiffened cylindrical shell shown
in Fig. 2 is approximately given by:
=
ms 2π R ρ s ( tL + nA )
(1)
Fig. 2: Cylindrical shell with ring stiffened
The following variables will be introduced when 𝑡𝑡0
represents the thickness of unstiffened cylindrical shell:
t
A
=
,y
t0
t0 L
(2)
Y
=
x
Introducing variables (2) into Eq. (1) gives, after
reordering:
x + ny − 1 =
0
(3)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
n=1
n=2
n=4
n=8
0
Solving this quadratic equation the following
formula for variable n arises:
n=
(1 − x ) / y
(4)
As it was mentioned above the axial length L of the
shells and the radius R are constant. And then, it is clear
that changing parameters x and y, a series lines
represented shells with constant mass can be got. An
example is shown in Fig. 3.
The value of x and y obtained from the Eq. (3) and
(4) provided the stiffened cylindrical shell with the
considered mass. In Fig. 3, the stiffened cylindrical shells
are the intersection points of the constant mass lines with
the coordinate axis. Summing up, it is clear now that by
assuming the value of x and y, the other parameters of the
shell is obtained.
From the Fig. 3, the point (x = 1 and y = 0)
represents an un-stiffened cylindrical shell. The
intersection points of each line with the axis of
coordinates represent stiffened cylindrical shells. Besides
these points, others are corresponding to stiffened
cylindrical shells with different geometry. Considering
the longitudinal coordinate, all points in this line
represent illegal parameters of shells. So it is not need to
be considered here.
An example will be shown now. As a reference an
un-stiffened cylindrical shell of the mass is chosen. The
0.1
0.2
0.3 0.4 0.5
X
0.6
0.7
0.8
0.9 1.0
Fig. 3: A family of constant mass lines for different ribs
Table 1: A family of shells with constant mass
𝑚𝑚𝑠𝑠 = 1183.2 kg L = 2000 mm 𝑡𝑡0 = 20 mm R = 600 mm
Constant values
--------------------------------------------------------------------Parameter
n=1
n=2
n=3
n=4
2
a A (mm )
4000
2000
1000
500
t (mm)
18
18
18
18
b A (mm2 )
8000
4000
2000
1000
t (mm)
16
16
16
16
2
c A (mm )
12000
6000
3000
1500
t (mm)
14
14
14
14
d A (mm2 )
16000
8000
4000
2000
t (mm)
12
12
12
12
e A (mm2 )
20000
10000
5000
2500
t (mm)
10
10
10
10
f
A (mm2 )
24000
12000
6000
3000
t (mm)
8
8
8
8
g A (mm2 )
28000
14000
7000
3500
t (mm)
6
6
6
6
radius of the shell is R = 600 mm and the thickness is
𝑡𝑡0 = 20 mm. This determines the value L = 2000 mm.
And the properties of the material are:
E=
2.05 × 105 MPa,υ =
0.3, ρ =
7.85 g / cm3
Now by choosing arbitrary values of the number of
ribs n and calculating values of parameters A and t by
expressions (2) and (4), it is possible to got a family of
stiffened cylindrical shells which evolved from keeping
constant mass. These shells are shown in Table 1.
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Res. J. Appl. Sci. Eng. Technol., 6(15): 2757-2763, 2013
FEM ANALYSES
A buckling analysis has been carried out in order to
show how the material of the cylindrical shell influences
the critical load under uniform external pressure. So a
family of stiffened cylindrical shells with constant mass
has been considered. The analysis was carried out by
ANSYS procedure. A Solid 45 finite solid element has
been chosen with eight nodes and six degrees of freedom
in each node. The boundary conditions imposed in the
analysis are shown in Fig. 4. The movement of a normal
edge direction and a rotation is allowed. A shell is loaded
with uniform external pressure only. Because the applied
pressure value equals unity (p = 1 MPa) and no preload
(dead load) was applied, the eigenvalue obtained from
the analysis is corresponds to the buckling load directly.
The material specification is the same as in the examples
above, it is:
Fig. 4: Boundary conditions
E=
2.05 ×105 MPa,υ =
0.3, ρ =
7.85 g / cm3
The first step, a mesh convergence analysis was
made by un-stiffened cylindrical shell under the same
condition. Figure 5 shows the buckling shape of the unstiffened cylindrical shell as an example. The results are
Fig. 5: Buckling shape of unstiffened cylindrical shell
(a) Change of load
(b) Relative change of load
Fig. 6: Mesh convergence analysis for unstiffened cylindrical shell
140
Relative change of Pcr (%)
0.08
Pcr (MPa)
120
100
80
60
40
20
4000
8000 12000 16000 20000 24000 28000
Number of elements
32000
(a) Change of load
0.07
0.06
0.05
0.04
0.03
0.02
5000
10000 15000
20000 25000 30000
Number of elements
(b) Relative change of load
Fig. 7: Mesh convergence analysis for stiffened cylindrical shell
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Res. J. Appl. Sci. Eng. Technol., 6(15): 2757-2763, 2013
POLYNOMIAL FUNCTIONS
t = 10 mm
50
t = 12 mm
t = 16 mm
70
60
t = 14 mm
Different shell thickness
t = 18 mm
Critical load (MPa)
90
80
40
30
20
10
0
2
1
3
4
5
Number of ribs
6
7
8
Fig. 8: Parts of the FEM analysis results
shown in Fig. 6. It was decided that 30,000 finite
elements of the shell would be enough for the analysis.
And the analysis results of f8 of the stiffened cylindrical
shell also present 30,000 finite elements showed in
Fig. 7.
The second step, the family of stiffened cylindrical
shells with constant mass was examined. The FEM
analysis results are shown in Fig. 8. The a4 of stiffened
cylindrical shell is corresponds to the critical load
obtained from ANSYS, which equals p cr = 37.782 MPa.
Similar value, that is, p cr = 36.254 MPa, gives by
comprehensive consideration of the Eq. (5) and (6).
The critical load of local buckling:
p=
0.6 Et 2
u − 0.37 R 2
(5)
The buckling load is one of the most important
objectively design variable of an externally pressurized
stiffened cylindrical shell. It integrates structural
parameters which react the bearing capacity. The
relationship between the structure parameters and
buckling load is nonlinear strongly. The simulation
results are not identical with the real value and the
relationship between material distribution and buckling
load cannot be got from Eq. (5) and (6) directly. So, a
similar polynomial function which reflects the changing
trend of these discrete data points needs to be got by
surface fitting.
Surface fitting is a data processing method, which
uses a continuous surface to depict the functional relation
among space discrete points. It can be constructed by the
least square method which does not need to go through
all the data points. Recently, moving least square method
is proposed for fitting the data. Compared with the
traditional least square method, it has obvious advantages
and is used broad in the data fitting and analysis. In this
study, the fitting surface needs to go through one key
point, so the interpolation conditions of the moving least
square method should be considered.
According to the paper of Feng-Mei et al. (2011), in
the moving least square method, the objective
polynomial function can be expressed as:
=
f ( x)
m
α ( x) p ( x)
∑=
i =1
And the critical pressure of general buckling is given by:
 α

 2 2 2+

 (α + n )
t
E
p= 2


2
n − 1 + 0.5α 2 1.68
2 R
βt
2
n
1
−
(
)


3
 αU (1 + β ) R

i
i
pT ( x )α ( x )
(7)
The coefficients are:
4
α ( x ) = α1 ( x ) ,α 2 ( x ) ,...,α m ( x ) 
(6)
The basic function is:
The non-dimensional parameters relating to the
design variables and constants are defined according to
theory and experiments as follows:
u=
T
0.6427l
Rt
p ( x ) =  p1 ( x ) , p2 ( x ) ,..., pm ( x ) 
Considering the complete polynomial basis, such as:
Linear base:
=
p( x)
β = lt / F
T
x, y ) ( m
(1,=
T
3)
Quadratic base:
α =πR / L
=
p( x)
(
=
U 0.535Lt 3 / I R ( t + F / l ) / l
)
And it can verify that parameters m = 2, n = 6 in
expression (6) gives the smallest value.
x, y, x , xy, y ) ( m 6 )
(1,=
2
2 T
In order to obtain a more accurately local
approximation, the difference of the weighted square
between local approximations f (x i ) and point value
should be minimum. In the paper of Xiao-Hong et al.
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Res. J. Appl. Sci. Eng. Technol., 6(15): 2757-2763, 2013
(2005) and Ni et al. (2010), the discretely weighted norm
of residual is:
n
J=
∑ w ( x − xi )  pT ( xi )α ( x ) − yi 
2
so, the objective function is substituted by Eq. (12)
through (11):
=
f ( x)
(8)
i =1
i =1
w (x-𝑥𝑥𝑖𝑖 ) is the weight function of the points x i while n is
the number of points in the solving domain and f (x) is
the fitting function.
Usually, the circle is chosen as the support domain
the spline function is constructed to be the weight
function.
Let s' = x-x i , s = s'/r, then the cubic spline function
is:
1
2
2
3
 3 − 4s + 4s , s ≤ 2

4 1
4
w ( s ) =  − 4s + 4s 2 − s 3 , < s < 1
3 2
3
0, s > 1


m
=
φ y
∑
(9)
k
i
ψ k ( x )Y
i
(12)
where, Ψ 𝑘𝑘 (x) is the shape function and k is the order of
basic function:
=
ψ k ( x ) =
φ1k ,φ2k ,...,φnk  pT ( x ) A−1 ( x ) B ( x )
(13)
While the fitting function should go through some
key points, the moving least square method with
interpolation conditions is applied to construct the fitting
surface formula.
Supposing there are a set of scattered nodes
(𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖 , 𝑧𝑧𝑖𝑖 ), i = 1, 2,.., n and the interpolation conditions
are (𝑥𝑥𝑠𝑠 , 𝑦𝑦𝑠𝑠 , 𝑧𝑧𝑠𝑠 ), s = 1, 2,.., t, t≤n. P (x, y) is the fitting
surface constructed by the moving least square method,
then, the fitting surface with interpolation conditions can
be written as:
t
when J takes the minimum value, matrix form of Eq. (8)
is as follows:
J=
( Pa ( x ) − Y ) W ( x ) ( Pa ( x ) − Y )
T
=
z P ( x, y ) − ∑ ls ( x, y ) δ s
where,
(10)
=
ls ( x )
where,
Y = ( y1 , y2 ,..., yn )
T
•
•
•
a (x) is obtained by the least square method:
B ( x ) = PTW ( x )
j =1
s≠ j
j
s
j
j
s
j
This formula can also be considered as the
modification of the moving least square fitting formula
where the correction function is ∑𝑡𝑡𝑠𝑠=1 𝑙𝑙𝑠𝑠 (x) 𝛿𝛿𝑥𝑥 .
So, in this study, the main processes of fitting
function are:
 p1 ( x1 ) p2 ( x1 ) ... pm ( x1 ) 


p (x )

P= 1 2




... pm ( xn ) 
 p1 ( xn )
A ( x ) = PTW ( x ) P
t
δ s P ( xs , y s ) − z s
=
=
wi w ( x − xi )
where,
(x − x ) ( y − y )
∏(x − x ) ⋅ ( y − y )
s = 1,2,..., t
W ( x ) = diag ( w1 ( x ) , w2 ( x ) ,..., wn ( x ) )
a ( x ) = A−1 ( x ) B ( x ) Y
(14)
s =1
Obtain the moving least square fitting surface
without interpolation conditions.
Calculate the deviation of the key points.
Obtain the moving least square fitting surface with
interpolation conditions.
According to the simulation results, the sets of
scattered nodes are got. In order to analysis the material
utilization, the dimensionless values should be
independent variables, the buckling load values should
be dependent variables. The detailed data is shown in
Table 2.
The MATLAB simulation tool was used to write the
fitting procedures and paint the surface graph. The fitting
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Res. J. Appl. Sci. Eng. Technol., 6(15): 2757-2763, 2013
Fig. 10: Surface of X partial derivative function
Fig. 9: The fitting surface
Table 2: The original data of surface fitting
z
z
x
y
20.414
0.9
0.1000
11.508
33.423
0.9
0.0500
18.631
37.782
0.9
0.0250
37.092
26.533
0.9
0.0125
84.627
15.568
0.8
0.2000
7.871
25.572
0.8
0.1000
12.644
51.306
0.8
0.0500
25.620
5.059
0.5
0.5000
54.605
8.230
0.5
0.2500
12.648
16.594
0.5
0.1250
52.306
0.8
0.0250
83.849
Table 3: The relative error of fitting surfaces
The maximal error
The minimal error
13%
0.6%
x
0.7
0.7
0.7
0.7
0.6
0.6
0.6
0.5
1
y
0.3000
0.1500
0.0750
0.0375
0.4000
0.2000
0.1000
0.0625
0
0.6
0.0500
The average error
2.6%
Fig. 11: Surface of Y partial derivative function
surface was constructed by the interpolation nodes which
are (12.648, 1, 0), which is shown in Fig. 9 and the
formula of fitting is expressed as:
3
2
1
i 4 −i
i 3−i
i 2 −i
i
i
i
=i 0 =i 0=i 0 =i 0
f ( x, y ) =
4
∑a x y
+ ∑b x y
+ ∑c x y
+ ∑ di xi y i −i + e
where,
 a0

 b0
 c0

d0
 e
a1 a2 a3 a4   −8230 −73500 −55830 136760 123180 
 

0
b1 b2 b3
−190660
  30030 164170

 =  −39130 −107030 53670

c1 c2
 

d1
  21120 16490

  −3770

 
According to the fitting function, the relative errors
among fitting surfaces and original data are shown in
Table 3. The error fits for the accuracy requirement.
MATERIAL UTILIZATION
Maximizing material utilization in structure design
is paramount important. Materials typically represent
75% and even more of the total costs in manufacture, so
a poor design can significantly increase costs of structure
over its life. The concept of material utilization is a
common vocabulary in energy industry, materials
science and etc., but relatively new in structure design.
The definition is also different from different structure
design. In this study, Material utilization is defined as the
effective degree of material use in stiffened cylindrical
shells structure. With the same material amount the
higher the material utilization ratio, the stronger the
carrying capacity of structure. So, the material utilization
is the change rate of the function of structure parameters
and target variable.
In the fitting function, the independent variables are
the shell and rib proportions of the total material.
Therefore, the effective degree of material usage is the
main factor that the independent variables affect the
dependent variable. Partial derivative function for X and
Y coordinate axis are got respectively. As shown in
Fig. 10 and 11, the change of the parameter x is higher
than the change of the parameter y. That is to say, in this
case, the utilization of the shell is higher than the
utilization of the rib.
Material utilization will be the new criteria and
method of structural design. Through the procedures, the
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Res. J. Appl. Sci. Eng. Technol., 6(15): 2757-2763, 2013
utilization of the shell and ribs in other cases can be got.
How to measure material utilization and its usage in
structures design are the key contents of the authors’
research in the future.
CONCLUSION
A simple procedure to create a family of stiffened
cylindrical shells, which is the revolution of constant
mass, has been shown. A lot of data points about
buckling load and material shares of the shell and ribs are
got form buckling analysis using ANSYS code. A
conclusion can be drawn from the fitting function: in the
case of uniform external pressure load stiffened
cylindrical shells, the utilization of the shell are higher
than the rib. Material utilization in structural system has
special significance to design reasonable structures and
ensure the safety of the overall structure. In the future
study, the criteria of material utilization in structural
design will be studied based on the results of this study.
This research project is innovative in its approach of
stiffened cylindrical shells design. It is expected that this
approach will make immense contribution to the
structural design.
ACKNOWLEDGMENT
The authors would like to express their gratitude and
sincere appreciation to the anonymous reviewers for
constructive suggestions to improve the quality and
readability of this study. This study was conducted with
supported of the Defense Pre-research Foundation of
Shipbuilding Industry (No. 11J1.3.1).
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